2. Prandtl – Meyer expansion wave 10/11/2021 1 Special thanks from Professor Mofid Gorji Presented at NIT

3. 10/11/2021 2 Ludwig Prandtl (4 February 1875 – 15 August 1953) was a German fluid dynamicist, physicist and aerospace scientist. He was a pioneer in the development of rigorous systematic mathematical analyses which he used for underlying the science of aerodynamics, which have come to form the basis of the applied science of aeronautical engineering. In the 1920s he developed the mathematical basis for the fundamental principles of subsonic aerodynamics in particular and in general up to and including transonic velocities. His studies identified the boundary layer, thin-airfoils, and lifting-line theories. The Prandtl number was named after him. Prandtl and his student Theodor Meyer developed the first theories of supersonic shock waves and flow in 1908. The Prandtl– Meyer expansion fans allowed for the construction of supersonic wind tunnels. He had little time to work on the problem further until the 1920s, when he worked with Adolf Busemann and created a method for designing a supersonic nozzle in 1929. Today, all supersonic wind tunnels and rocket nozzles are designed using the same method. A full development of supersonics would have to wait for the work of Theodore von Kármán, a student of Prandtl at Göttingen. Prandtl worked at Göttingen until he died on 15 August 1953. His work in fluid dynamics is still used today in many areas of aerodynamics and chemical engineering. He is often referred to as the father of modern aerodynamics.

4. 10/11/2021 3 Theodor Meyer (July 1, 1882 – March 8, 1972 in Bad Bevensen, Germany) was a mathematician, a student of Ludwig Prandtl, and a founder of the scientific discipline now known as compressible flow or gas dynamics. After the war, Meyer sought further employment in theoretical physics but could not find it in depression-era postwar Germany. Ludwig Prandtl was not financially able to hire him, but Meyer did design the de Laval nozzle for a supersonic wind tunnel that Prandtl wanted to build.Prandtl sought funding from the German military to build this advanced aerodynamic test facility, but he did not succeed. Meyer subsequently worked as an engineer and as a high-school teacher of math and physics. By the time of his death at almost age 90 in 1972, not even his family or his neighbors in Bad Bevensen, Germany were aware of the formative role he had played, with Ludwig Prandtl, in the scientific discipline known as compressible flow or gas dynamics. Until recently, nothing was known about Theodor Meyer's life after he finished his Ph.D. research in 1908. We now know that he served as a junior officer in the German infantry during World War I.He was injured in trench warfare on the infamous Western Front, and he came into contact with Fritz Haber, later a Nobel Prizewinner and now known as the "father of chemical warfare."

5. As an object moves through a gas, the gas molecules are deflected around the object. If the speed of the object is much less than the speed of sound of the gas, the density of the gas remains constant and the flow of gas can be described by conserving momentum, and energy. As the speed of the object approaches the speed of sound, we must consider compressibility effects on the gas. The density of the gas varies locally as the gas is compressed by the object. For compressible flows with little or small flow turning, the flow process is reversible and the entropy is constant. The change in flow properties are then given by the isentropic relations (isentropic means "constant entropy"). But when an object moves faster than the speed of sound, and there is an abrupt decrease in the flow area, shock waves are generated in the flow. Shock waves are very small regions in the gas where the gas properties change by a large amount. Across a shock wave, the static pressure, temperature, and gas density increases almost instantaneously. The changes in the flow properties are irreversible and the entropy of the entire system increases. Because a shock wave does no work, and there is no heat addition, the total enthalpy and the total temperature are constant. 10/11/2021 4 Shock wave

6. But because the flow is non-isentropic, the total pressure downstream of the shock is always less than the total pressure upstream of the shock. There is a loss of total pressure associated with a shock wave as shown on the slide. Because total pressure changes across the shock, we can not use the usual (incompressible) form of Bernoulli's equation across the shock. The Mach number and speed of the flow also decrease across a shock wave. If the shock wave is perpendicular to the flow direction, it is called a normal shock. There are equations which describe the change in the flow variables. The equations are derived from the conservation of mass, momentum, and energy. Depending on the shape of the object and the speed of the flow, the shock wave may be inclined to the flow direction. When a shock wave is inclined to the flow direction it is called an oblique shock. On this slide we have listed the equations which describe the change in flow variables for flow across an oblique shock. The equations presented here were derived by considering the conservation of mass, momentum, and energy for a compressible gas while ignoring viscous effects. The equations have been further specialized for a two-dimensional flow without heat addition. The equations only apply for those combinations of free stream Mach number and deflection angle for which an oblique shock occurs. If the deflection is too high, or the Mach too low, a normal shock occurs. For the Mach number change across an oblique shock there are two possible solutions; one supersonic and one subsonic. In nature, the supersonic ("weak shock") solution occurs most often. However, under some conditions the "strong shock", subsonic solution is possible. 10/11/2021 5

7. Oblique shocks are generated by the nose and by the leading edge of the wing and tail of a supersonic aircraft. Oblique shocks are also generated at the trailing edges of the aircraft as the flow is brought back to free stream conditions. Oblique shocks also occur downstream of a nozzle if the expanded pressure is different from free stream conditions. In high speed inlets, oblique shocks are used to compress the air going into the engine. The air pressure is increased without using any rotating machinery. On the slide, a supersonic flow at Mach number M approaches a shock wave which is inclined at angle s. The flow is deflected through the shock by an amount specified as the deflection angle - a. The deflection angle is determined by resolving the incoming flow velocity into components parallel and perpendicular to the shock wave. The component parallel to the shock is assumed to remain constant across the shock, the component perpendicular is assumed to decrease by the normal shock relations. 10/11/2021 6

8. 7 What are SHOCK WAVES Normal shock wave Types of Shock waves

9. Combining the components downstream of the shock determines the deflection angle. Then: cot(a) = tan(s) * [{((gam+1) * M^2)/(2 * M^2 * sin^2(s) - 1)} - 1] where tan is the trigonometric tangent function, cot is the co-tangent function: cot(a) = tan(90 degrees - a) and sin^2 is the square of the sine. Gam is the ratio of specific heats. Across the shock wave the Mach number decreases to a value specified as M1: M1^2 * sin^2(s -a) = [(gam-1)M^2 sin^2(s) + 2] / [2 * gam * M^2 * sin^2(s) - (gam - 1)] The right hand side of all these equations depend only on the free stream Mach number and the shock angle. The shock angle depends in a complex way on the free stream Mach number and the wedge angle. So knowing the Mach number and the wedge angle, we can determine all the conditions associated with the oblique shock. The equations describing oblique shocks were published in NACA report (NACA-1135) in 1951. 10/11/2021 8

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11. A supersonic expansion fan, technically known as Prandtl–Meyer expansion fan, a two-dimensional simple wave, is a centered expansion process that occurs when a supersonic flow turns around a convex corner. The fan consists of an infinite number of Mach waves, diverging from a sharp corner. When a flow turns around a smooth and circular corner, these waves can be extended backwards to meet at a point. Each wave in the expansion fan turns the flow gradually (in small steps). It is physically impossible for the flow to turn through a single "shock" wave because this would violate the second law of thermodynamics. Across the expansion fan, the flow accelerates (velocity increases) and the Mach number increases, while the static pressure, temperature and density decrease. Since the process is isentropic, the stagnation properties (e.g. the total pressure and total temperature) remain constant across the fan. The theory was described by Theodor Meyer on his thesis dissertation in 1908, along with his advisor Ludwig Prandtl, who had already discussed the problem a year before. 10/11/2021 10

12. As Mach number varies from 1 to inf, v takes values from 0 to V max, where This places a limit on how much a supersonic flow can turn through, with the maximum turn angle given by, One can also look at it as follows. A flow has to turn so that it can satisfy the boundary conditions. In an ideal flow, there are two kinds of boundary condition that the flow has to satisfy, 1.Velocity boundary condition, which dictates that the component of the flow velocity normal to the wall be zero. It is also known as no-penetration boundary condition. 2.Pressure boundary condition, which states that there cannot be a discontinuity in the static pressure inside the flow (since there are no shocks in the flow). If the flow turns enough so that it becomes parallel to the wall, we do not need to worry about pressure boundary condition. However, as the flow turns, its static pressure decreases (as described earlier). If there is not enough pressure to start with, the flow won't be able to complete the turn and will not be parallel to the wall. This shows up as the maximum angle through which a flow can turn. The lower the Mach number is to start with (i.e. small M1 ), the greater the maximum angle through which the flow can turn. 10/11/2021 11

13. The streamline which separates the final flow direction and the wall is known as a slipstream (shown as the dashed line in the figure(Slide 7)). Across this line there is a jump in the temperature, density and tangential component of the velocity (normal component being zero). Beyond the slipstream the flow is stagnant (which automatically satisfies the velocity boundary condition at the wall). In case of real flow, a shear layer is observed instead of a slipstream, because of the additional no-slip boundary condition. As an object moves through a gas, the gas molecules are deflected around the object. If the speed of the object is much less than the speed of sound of the gas, the density of the gas remains constant and the flow of gas can be described by conserving momentum and energy. As the speed of the objects increases towards the speed of sound, we must consider compressibility effects on the gas. The density of the gas varies locally as the gas is compressed by the object. When an object moves faster than the speed of sound, and there is an abrupt decrease in the flow area, shock waves are generated. If the flow area increases, however, a different flow phenomenon is observed. If the increase is abrupt, we encounter a centered expansion fan. 10/11/2021 12

14. 10/11/2021 13 Gas dynamics-Prandtl-Meyer flow

15. There are some marked differences between shock waves and expansion fans. Across a shock wave, the Mach number decreases, the static pressure increases, and there is a loss of total pressure because the process is irreversible. Through an expansion fan, the Mach number increases, the static pressure decreases and the total pressure remains constant. Expansion fans are isentropic. The calculation of the expansion fan involves the use of the Prandtl-Meyer function. This function is derived from conservation of mass, momentum, and energy for very small (differential) deflections. The Prandtl-Meyer function is denoted by the Greek letter nu on the slide and is a function of the Mach number M and the ratio of specific heats gam of the gas. nu = {sqrt[(gam+1)/(gam-1)]} * atan{sqrt[(gam-1)*(M^2 - 1)/(gam+1)]} - atan{sqrt[M^2 -1]} where atan is the trigonometric inverse tangent function. It is also written as shown on the slide tan^-1. The meaning of atan can be explained by these two equations: atan(a) = b tan(b) = a 10/11/2021 14

16. As mentioned above, the Mach number of a supersonic flow increases through an expansion fan. The amount of the increase depends on the incoming Mach number and the angle of the expansion. The physical interpretation of the Prandtl-Meyer function is that it is the angle through which you must expand a sonic (M=1) flow to obtain a given Mach number. The value of the Prandtl-Meyer function is therefore called the Prandtl-Meyer angle. 10/11/2021 15

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21. 10/11/2021 20 Thanks for your attention